A208435 Expansion of x * f(x) * f(-x^12)^3 * psi(x^3) / psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.

0, 1, 1, -2, 0, 3, -2, -4, 0, 5, 1, -8, 0, 7, -4, -8, 0, 9, 8, -10, 0, 14, -6, -12, 0, 16, 6, -14, 0, 15, -8, -20, 0, 17, 14, -18, 0, 19, -10, -24, 0, 26, 1, -22, 0, 23, -16, -28, 0, 25, 20, -32, 0, 32, -14, -28, 0, 29, 12, -30, 0, 38, -16, -32, 0, 33, 31

Offset

0,4

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-2/3) * eta(q^2)^4 * eta(q^6)^2 * eta(q^12)^3 / (eta(q) * eta(q^3) * eta(q^4)^3) in powers of q.

Euler transform of period 12 sequence [ 1, -3, 2, 0, 1, -4, 1, 0, 2, -3, 1, -4, ...].

a(4*n) = 0. 24 * a(n) = A207541(3*n + 2).

Example

x + x^2 - 2*x^3 + 3*x^5 - 2*x^6 - 4*x^7 + 5*x^9 + x^10 - 8*x^11 + ...
q^5 + q^8 - 2*q^11 + 3*q^17 - 2*q^20 - 4*q^23 + 5*q^29 + q^32 - 8*q^35 + ...

Mathematica

a[n_]:=SeriesCoefficient[((QP[q^2]^4*QP[q^6]^2*QP[q^12]^3)/(QP[q]*QP[q^3]*
QP[q^4]^3)), {q, 0, n}]; Table[a[n], {n, -1, 50}] (* G. C. Greubel, Dec 17 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^3), n))}

Crossrefs

Cf. A207541.

Sequence in context: A213607 A298932 A089196 * A208457 A353335 A232343

Adjacent sequences: A208432 A208433 A208434 * A208436 A208437 A208438

Keyword

sign

Author

Michael Somos, Feb 26 2012

Status

approved